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Plastic notch stress intensity factors for pointed V-notches under antiplane shear loading

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Abstract

The paper deals with a work-hardening, elastic–plastic, stress analysis of pointed V-notches under antiplane shear deformation loading both under small and large scale yielding. Stress and strain field intensities are expressed in terms of plastic Notch Stress Intensity Factors, which are analytically linked to the corresponding linear elastic ones under small scale yielding. The near tip stress and strain fields are then used to give closed-form expressions for the Strain Energy Density over a circular sector surrounding the notch tip, and for the J-integral parameter, both as a function of the relevant plastic NSIFs, these expressions being valid both under small and large scale yielding.

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Abbreviations

K 3 :

Non linear notch stress intensity factor

K 3,e :

Elastic notch stress intensity factor

K 3,p :

Plastic notch stress intensity factor

r p :

Plastic radius as estimated through the linear elastic stress fields

\({R\left({\overline{\varphi}} \right)}\) :

Equation for the stress isolines in the plastic region

\({R_{\rm p}\left({\overline{\varphi}}\right)}\) :

Equation for the elastic–plastic boundary as estimated through the plastic stress fields

x p :

Extension of the plastic zone on the notch bisector as estimated through the plastic stress fields

Δp :

x p − r p

C p :

Amplification factor due to stress redistribution in Irwin’s approach

J 3,p :

Plastic J integral under mode III

\({{J}_{3,{\rm p}}^{\rm L}}\) :

Normalised plastic J integral

B J :

Parameter useful for evaluating J 3,p

J III :

Elastic J integral for the crack case

J 3,e :

Elastic J integral

\({{J}_{3,{\rm e}}^{\rm L}}\) :

Normalised elastic J integral

E p :

Total strain energy density in control volume under plastic conditions

\({\overline{{W}}_{\rm p}}\) :

Plastic strain energy density averaged on the control volume

B W :

Parameter useful for evaluating \({\overline{{W}}_{\rm p}}\)

\({\overline{{W}}_{\rm e}}\) :

Elastic strain energy density averaged on the control volume

References

  • Berto F, Lazzarin P, Matvienko YG (2007) J-integral evaluation for U- and V-blunt notches under mode I loading and materials obeying a power hardening law. Int J Fract 146: 33–51. doi:10.1007/s10704-007-9134-x

    Article  Google Scholar 

  • Chen YH, Hasebe N (1997) Explicit formulations of the J-integral considering higher order singular terms in eigenfunction expansion forms Part I. Analytical treatments. Int J Fract 85: 11–34. doi:10.1023/A:1007486727751

    Article  Google Scholar 

  • Chen DH, Ushijima K (2000) Plastic stress singularity near the tip of a V-notch. Int J Fract 106: 117–134. doi:10.1023/A:1007693716844

    Article  Google Scholar 

  • Filippi S, Ciavarella M, Lazzarin P (2002) An approximate, analytical approach to the ‘HRR’-solution for sharp V-notches. Int J Fract 117: 269–286. doi:10.1023/A:1022057621185

    Article  Google Scholar 

  • Gdoutos EE (1990) Fracture mechanics criteria and applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Gross R, Mendelson A (1972) Plane elastostatic analysis of V-notched plates. Int J Fract Mech 8: 267–272. doi:10.1007/BF00186126

    Article  Google Scholar 

  • Hui CY, Ruina A (1995) Why K? Higher order singularities and small scale yielding. Int J Fract 72:97–120

    Google Scholar 

  • Hult JAH (1957) Elastic–plastic torsion of sharply notched bars. J Mech Phys Solids 6: 79–82. doi:10.1016/0022-5096(57)90050-9

    Article  Google Scholar 

  • Hult JAH, McClintock FA (1956) Elastic–plastic stress and strain distribution around sharp notches under repeated shear. In: 9th Int Cong Appl Mech, 8, Brussels

  • Hutchinson JW (1968) Singular behavior at the end of a tensile crack in a hardening material. J Mech Phys Solids 16: 13–31. doi:10.1016/0022-5096(68)90014-8

    Article  Google Scholar 

  • Hutchinson JW (1968) Plastic stress and strain fields at a crack tip. J Mech Phys Solids 16: 337–347. doi:10.1016/0022-5096(68)90021-5

    Article  Google Scholar 

  • Koskinen MF (1963) Elastic–plastic deformation of a single grooved flat under longitudinal shear. J Basic Eng 85: 585–594

    Google Scholar 

  • Kuang ZB, Xu XP (1987) Stress and strain fields at the tip of a sharp V-notch in a power hardening material Int J Fract 35: 39–53

    Google Scholar 

  • Larsson SG, Carlsson AJ (1973) Influence of nonsingular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 21: 263–277

    Article  Google Scholar 

  • Lazzarin P, Zambardi R (2001) A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. Int J Fract 112: 275–298

    Article  Google Scholar 

  • Lazzarin P, Berto F (2008) Control volumes and strain energy density under small and large scale yielding due to tension and torsion loading. Fatigue Fract Eng Mater Struct 31: 95–107

    Google Scholar 

  • Lazzarin P, Zambardi R (2002) The equivalent strain energy density approach re-formulated and applied to sharp V-shaped notches under localized and generalized plasticity. Fatigue Fract Eng Mater Struct 25: 917–928

    Article  Google Scholar 

  • Lazzarin P, Zambardi R, Livieri P (2001) Plastic notch stress intensity factors for large V-shaped notches under mixed load conditions. Int J Fract 107: 361–377

    Article  Google Scholar 

  • Lazzarin P, Livieri P, Zambardi R (2002) A J-integral-based approach to predict the fatigue strength of components weakened by sharp V-shaped notches. Int J Comp Appl Technol 15(4/5): 202–210

    Article  Google Scholar 

  • Lazzarin P, Sonsino CM, Zambardi R (2004) A notch stress intensity approach to predict the fatigue behaviour of T butt welds between tube and flange when subjected to in–phase bending and torsion loading. Fatigue Fract Eng Mater Struct 27: 127–141

    Article  CAS  Google Scholar 

  • Livieri P (2008) Use of J-integral to predict static failures in sharp V-notches and rounded U-notches. Eng Fract Mech 75: 1779–1793

    Article  Google Scholar 

  • Molski K, Glinka G (1981) A method of elastic–plastic stress and strain calculation at a notch root. Mater Sci Eng 50: 93–100

    Article  CAS  Google Scholar 

  • Neuber H (1958) Theory of notch stresses. Springer-Verlag, Berlin

    Google Scholar 

  • Neuber H (1968) Über die Berücksichtigung der Spannungskonzentration bei Festigkeitsberechnungen. Konstruktion 20: 245–251

    Google Scholar 

  • Rice JR (1966) Contained plastic deformation near cracks and notches under longitudinal shear. Int J Fract Mech 2: 426–447

    CAS  Google Scholar 

  • Rice JR (1967) Stresses due to a sharp notch in a work-hardening elastic–plastic material loaded by longitudinal shear. J Appl Mech 34: 287–298

    Google Scholar 

  • (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35: 379–386

    Google Scholar 

  • Rice JR (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22: 17–26

    Article  Google Scholar 

  • Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16: 1–12

    Article  Google Scholar 

  • Sharma SM, Aravas N (1991) Determination of higher-order terms in asymptotic elastoplastic crack tip solutions. J Mech Phys Solids 39: 1043–1072

    Article  Google Scholar 

  • Szanto M, Read DT (1992) J-integral applied to stress-strain concentrations. Eng Fract Mech 43: 401–415

    Article  Google Scholar 

  • Unger DJ (2001) Analytical fracture mechanics. Dover publication

  • Wang TJ, Kuang ZB (1999) Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading. Int J Fract 96: 303–329

    Article  Google Scholar 

  • Xia L, Wang T (1993) Singular behavior near the tip of a sharp V-notch in a power-law hardening material. Int J Fract 59: 83–93

    Google Scholar 

  • Xia L, Wang T, Shih CF (1993) Higher-order analysis of crack tip fields in elastic power-law hardening materials. J Mech Phys Solids 41: 665–687

    Article  Google Scholar 

  • Yang S, Yuan FG, Cai X (1996) Higher order asymptotic elastic–plastic crack-tip fields under antiplane shear. Eng Fract Mech 54: 405–422

    Article  Google Scholar 

  • Yuan FG, Yang S (1994) Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear. Int J Fract 69: 1–26

    Article  Google Scholar 

  • Zappalorto M, Lazzarin P (2007) Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading. Int J Fract 148: 139–154

    Article  Google Scholar 

  • Zhang N, Joseph PF (1998) A nonlinear finite element eigenanalysis of singular plane stress fields in bimaterial wedges including complex eigenvalues. Int J Fract 90: 175–207

    Article  Google Scholar 

  • Zwillinger D (1989) Handobook of differential equations. Academic Press, San Diego

    Google Scholar 

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Authors and Affiliations

  1. Department of Management and Engineering, University of Padova, 36100, Vicenza, Italy

    P. Lazzarin & M. Zappalorto

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  1. P. Lazzarin
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  2. M. Zappalorto
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Correspondence to P. Lazzarin.

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Lazzarin, P., Zappalorto, M. Plastic notch stress intensity factors for pointed V-notches under antiplane shear loading. Int J Fract 152, 1–25 (2008). https://doi.org/10.1007/s10704-008-9260-0

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